Inverting Integer Numbers - Is There a Formula?

[FenixDragon]

hi... I was thinking... is there any formula that inverts int numbers? like 21 transforms into 12... I have found an algorithm that do this... but I want to know if exists any formula to it... thx...

 

[HallsofIvy]

I will assume that you know both the given number x and the number of digits in x, n.

integer function Reverse(integer x, y)
{
y= 0; //comment: we will add parts of the answer to y
for i=1 to n
{
k= x/10^{n-i};
//comment: this is computer "integer" arithmetic : 3/2= 1, not 1.5
//comment: mathematically, k= (x- (x mod 10^{n-i}))/10^{n-i}
//comment: if x= 251, for example, n= 3, i=1, 10^{n-1}= 10^2= 100
//comment: then k= 251/100= 2 or k= (251- (251 mod 100))/100= (251- 51)/100= 2
y=y+ k*10^{i-1}
x= x- k;
}
return y;
}

With x= 251, n= 1, the first time through the loop, i= 1, k= 251/10^(3-1)= 251/100= 2 so y= 0+ 2*10^0= 2, x= 251- 200. The second time through the loop, i= 2, x= 51 now, k= 51/10^(3-2)= 51/10= 5 so y= 2+ 5*10= 52, x= 51-50. The third and last time through the loop, i= 3, x= 1, k= 1/10^(3-3)= 1/1= 1 and y= 52+ 1*100= 152, 251 reversed.

If you are given only x, not the number of digits in x, then you can find it this way.

n= 0;
k= 1;
while x>= k do
{
n= n+1
k= k*10
}

For example, if x= 251 again, we first see that 251> 1 so n= 0+ 1= 1. k= 1*10= 10. Since 251> 10, n= 1+1= 2, k= 10*10= 100. Since 251> 100, n= 2+ 1= 3, k= 10*100= 1000. Since 251< 1000, the loop terminates with n= 3, the number of digits in 251.

 

[FenixDragon]

sorry... but isn't exactly what I wanted... I already know this algorithm... but I want to know if exists any math function that do this... thx...

 

[matt grime]

Yes, here is the function:

f(x)= "x written in reverse order of its digits".

That is a function. Actually, we had better call this f_10, otherwise it is not a well defined function on the integers: it depends on the choice of base, so let f_n(x)="x written in reverse order of its digits in base n".

I suspect your conflating the notion of function with a method of evaluating that function on some given input.

 

[3trQN]

The base is important of course, because the string of digits is defined by the base you count in.

In base 10, the numbers 11 and 9 are important for reversable strings, factors of 11 tend be palindromic provided each of the digits is no greater than the base n/2. The difference between two mirror strings can be factored by 3^3.

Thats about all i know.

21 - 12 = 9
42 - 24 = 18
31 - 13 = 18
321 - 123 = 198 = 2 * 99 = 2*11*9

Think about the repeating decimal 0.99999...

 

[shmoe]

Yes, the age old problem of what do you mean by "formula"?


I'd stick with the base 2 version of this function. You can then just turn the page upside down, or look in a mirror.

 

[matt grime]

In base 10, the numbers 11 and 9 are important for reversable strings, factors of 11 tend be palindromic

The factors of 11 are 1 and 11 so they are all palindromic... (Yes, I know you meant multiple.)

 

[3trQN]

oops, well spotted...

 

[gnomedt]

Would this work as a symbolic formula for R_b(n), giving the number formed by reversing the digits of n in base b, when n is an integer?

$$R_b(n) = \sum_{k=1}^{\left \lceil log_b n \right \rceil} \left \lfloor \frac{n-b^k\left \lfloor \frac{n}{b^k} \right \rfloor}{b^{k-1}} \right \rfloor b^{\left \lceil log_b n \right \rceil - k}$$

Definitely could be simplified.

 

[bomba923]

Would this work as a symbolic formula for R_b(n), giving the number formed by reversing the digits of n in base b, when n is an integer?

$$R_b(n) = \sum_{k=1}^{\left \lceil log_b n \right \rceil} \left \lfloor \frac{n-b^k\left \lfloor \frac{n}{b^k} \right \rfloor}{b^{k-1}} \right \rfloor b^{\left \lceil log_b n \right \rceil - k}$$

Definitely could be simplified.

Your formula fails whenever n=b^m for any natural 'm'.*

*Hint: the number of digits of any $n \in \mathbb{N}$ in any base $b \in \mathbb{N} - \left\{ {1} \right\}$ is $\left\lfloor {\log _b n} \right\rfloor + 1$, not $\left\lceil {\log _b n}\right\rceil$
(since you probably don't want to exclude natural powers of 'b' from your formula ...)

---------------------------------------------------------
~Anyway, a simpler formula for R_b(n) is:

$$R_b (n) = \sum\limits_{i = 0}^{\left\lfloor {\log _b n} \right\rfloor } {\left( {\left\lfloor {\frac{n}{{b^i }}} \right\rfloor - b\left\lfloor {\frac{n}{{b^{i + 1} }}} \right\rfloor } \right)b^{\left\lfloor {\log _b n} \right\rfloor - i} }$$